Stable parallelizability of partially oriented flag manifolds (Q1075647)

This paper solves the questions of stable parallelizability and parallelizability for the family of partially oriented flag manifolds, except for a few undecided cases. In particular, for the oriented Grassmannians \(\tilde G_ k({\mathbb{R}}^ n)\) it is proved that apart from the spheres \(S^ 1,S^ 3\), and \(S^ 7\) only \(\tilde G_ 3({\mathbb{R}}^ 6)\) is parallelizable, and only \(\tilde G_ 2({\mathbb{R}}^ 4)\) is stably parallelizable and not parallelizable. Negative results are derived for the most part using KO theory and the ''inclusion method'', while positive results are mainly based on the ''\(\lambda\) \({}^ 2\) construction''.